It is well known that \( g(x, a)\)  can be considered as a density function of a weighted sum
of non-central \( \chi ^{2}\)  distributions.
An explicit but cumbersome representation for \( g(x, a) \)  in
finite dimensional space \({\bf H}\)  is available (see, e.g., Section 18 in
Johnson, Kotz and Balakrishnan [1]}).
However, it involves some special characteristics of the related
Gaussian measure which makes it hard to use
in specific situations.
Our result (REF )
is much more transparent and provide
sharp uniform upper bounds.
Indeed, in the case \({\bf H}= {\bf R}^{d}\) , \(a = 0\) , \(\Sigma \)  is the unit matrix, one has that
the distribution of \(||Z||^{2}\)  is the standard \(\chi ^{2}\)  with
\(d\)  degrees of freedom and the maximum of its probability density function is proportional to \(d^{-1/2}\) . This is the same as what we get in (REF ).
